#315787

order	512
nilpotency class	3
isoclinism	isoclinic class of #298795
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2856	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

722 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 722 conjugacy classes of subgroups — too many to draw. Only its 178 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2,4,6)(3,5,7,8)(9,10)(11,12)(13,15)(14,16)(17,18)(19,25)(20,29)(21,23)(22,27)(24,26)(28,31)(30,32);(1,3)(2,5)(4,7)(6,8)(10,15)(11,16)(17,19,20,27,21,28,26,32)(18,22,24,25,23,30,29,31);(3,7)(5,8)(9,11,12,10,13,16,14,15)(18,23)(22,30)(24,29)(25,31);(9,12,13,14)(10,16,15,11)(17,19,26,32,21,28,20,27)(18,22,29,31,23,30,24,25)

Polycyclic presentation:

2^9|a^2=gi,b^2=ei,c^2=fhi,d^2=eh,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=cg,d^b=dh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 222983);